Abstract

We prove a uniform boundedness principle for the Lipschitz seminorm of continuous, monotone, positively homogeneous, and subadditive mappings on suitable cones of functions. The result is applicable to several classes of classically nonlinear operators.

1. Introduction and Preliminaries

Uniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) is one of the cornerstones of classical functional analysis (see, e.g., [1–3] and the references cited therein). In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive, and Lipschitz mappings defined on a suitable cone of functions (Theorem 2). This result is applicable to several classes of classically nonlinear operators (Examples 4 and 5 and Remarks 7 and 8).

Let be a nonempty set. Throughout the article let denote a vector space of all functions or a vector space of all equivalence classes of (almost equal everywhere) real measurable functions on , if is a measure space. As usual, denotes the absolute value of .

Let be a vector space and let denote the positive cone of , that is, the set of all such that for all (almost all) . The space is called an ordered vector space with the partial ordering induced by the cone . If, in addition, is a normed space it is called an ordered normed space. The vector space is called a vector lattice (or a Riesz space) if for every we have a supremum and infimum (greatest lower bound) in . If, in addition, is a normed space and if implies , then is a called normed vector lattice (or a normed Riesz space). Note that in a normed vector lattice we have for all . A complete normed vector lattice is called a Banach lattice. Observe that itself is a vector lattice.

Let be a normed space. The cone is called normal if and only if there exists a constant such that whenever , . A cone is normal if and only if there exists an equivalent monotone norm on ; that is, whenever (see, e.g., [4, Theorem 2.38]). A positive cone of a normed vector lattice is closed and normal. Every closed cone in a finite dimensional Banach space is necessarily normal.

Let be a cone (not necessarily equal to ). A cone is said to be complete if it is a complete metric space in the topology induced by . In the case when is a Banach space this is equivalent to being closed in .

A mapping is called positively homogeneous (of degree 1) if for all and . A mapping is called Lipschitz if there exists such that for all and we denoteIf is Lipschitz and positively homogeneous, thenNote also that a Lipschitz and positively homogeneous mapping on is always bounded on ; that is, is finite and it holds that . Moreover, a positively homogeneous mapping , which is continuous at , is bounded on .

A set is called a wedge if and if for all . A wedge induces on a vector preordering ( if and only if ), which is reflexive and transitive, but not necessarily antisymmetric.

If is a wedge, then is called subadditive if for and is called monotone (order preserving) if whenever , . Note that in this definition of subadditivity and monotonicity we consider on (and on ) a partial ordering induced by (not a preordering ). One of the reasons for this choice is that, for example, it may happen that a nonlinear map is monotone with respect to the ordering , but it is not monotone with respect to the preordering (see, e.g., [5, Section ] and max-type operators, or [6] and the “renormalization operators” which occur in discussing diffusion on fractals). Moreover, for similar reasons wherever in our article we consider a subcone we consider on a partial ordering induced by (not a partial ordering ). Observe that in this setting the set is a vector subspace in and thus a wedge.

In our main result (Theorem 2) we will consider a normed space with a normal cone and a complete subcone that satisfies for all and such that for all , where . Since itself is a vector lattice the above assumptions make sense. Note also that a positive cone of each Banach lattice or, in particular, of each Banach function space (see, e.g., [2, 7–11] and the references cited therein) satisfies these properties. For the theory of cones, wedges, linear and nonlinear operators on cones and wedges, Banach ordered spaces, Banach function spaces, vector and Banach lattices, and applications, for example, in financial mathematics, we refer the reader to [2, 4, 5, 7, 8, 12–19] and the references cited therein.

2. Results

We will need the following lemma.

Lemma 1. Let be a vector space and let be a subcone such that for all . If is a subadditive and monotone mapping, thenfor all .If, in addition, is a normed space such that is normal and is a subcone such that for all , where , and if and is bounded on , then is Lipschitz on .

Proof. Let . Since is a subadditive, we haveIt follows that , since is monotone and . Similarly one obtains that , which proves (4).Assume that, in addition, is a normed space such that is normal (with a normality constant ) and a subcone such that for all and that and is bounded on . It follows from (4) thatand thus is Lipschitz on (and ), which completes the proof.

The following uniform boundedness principle is the central result of this article.

Theorem 2. Let be a normed space such that is normal and let be a complete subcone, such that for all and such that for all . Assume that is a set of subadditive and monotone mappings such that and that each is positively homogeneous and continuous on .If the set is bounded for each (i.e., for each there exists such that for all ), then there exists such that for all .

Proof. Since is closed and each is continuous on the setis closed in for each . Moreover, is a complete metric space and . By Baire’s theorem there exist , , and such that an open ball , where is the normality constant of .Let such that and . Since is a normal cone and is positively homogeneous on , we have by (4)Since is subadditive and monotone on we havewhich together with (8) impliesWe also haveIndeed, if , thenand if , thenwhich proves (11).It follows from (11) that and thus and so . Thereforeand so .

Remark 3. (i) Each satisfies (see the proof of Lemma 1). Therefore we could alternatively set in the proof above and prove a uniform upper bound for all , which gives the same conclusion.(ii) In the proofs of Lemma 1 and Theorem 2 we did not need the assumption , so it suffices to assume that is a wedge in these two results (not necessarily a cone).(iii) Also the assumption on normality of can be slightly weakened in Lemma 1 and Theorem 2. Instead of normality of it suffices to assume that there exists a constant such that whenever , (where again means ).

Our results can be applied to various classes of nonlinear operators. In particular, they apply to various max-kernel operators (and their isomorphic versions) appearing in the literature (see, e.g., [5, 19–21] and the references cited therein). We point out the following two related examples from [5, 17–19].

Example 4. Given , let be Banach lattice of continuous functions on equipped with norm. Consider the following max-type kernel operators of the formwhere and are given continuous functions satisfying . The kernel is a given nonnegative continuous function, where denotes the compact setIt is clear that for it holds that . The eigenproblem of these operators arises in the study of periodic solutions of a class of differential-delay equationswith state-dependent delay (see, e.g., [19]).The mapping is subadditive and monotone and is positively homogeneous and Lipschitz on . Moreover, , where . Clearly, Theorem 2 applies to sets of such mappings.Consequently, Theorem 2 applies also to isomorphic max-plus mappings (see, e.g., [19] and the references cited therein) and a Lipschitz seminorm with respect to a suitably induced metric. Note that a related result for uniform boundedness (in fact contractivity) result for a Lipschitz seminorm of semigroups of max-plus mappings was stated in [22, 23]. However, observe that the Lipschitz seminorm there is defined with respect to a different metric than that in our case.

Example 5. Let be a nonempty set and let be the set of all bounded real functions on . With the norm and natural operations, is a Banach lattice. Let and let satisfy . Let be defined by . Then is subadditive and monotone mapping that satisfies and is positively homogeneous and Lipschitz on ; therefore Theorem 2 applies to sets of such mappings. It also holds that . In particular, if is the set of all natural numbers , our results apply to infinite bounded nonnegative matrices (i.e., for all and ). In this case, and and .

Remark 6. The special case of Example 5 when for some is well known and studied under the name max-algebra (an analogue of linear algebra). Together with its isomorphic versions (max-plus algebra and min-plus algebra also known as tropical algebra) it provides an attractive way of describing a class of nonlinear problems appearing, for instance, in manufacturing and transportation scheduling, information technology, discrete event-dynamic systems, combinatorial optimization, mathematical physics, and DNA analysis (see, e.g., [24–29] and the references cited therein).

Remark 7. Our results apply also to more general max-type operators studied in [5, Section ]. The authors considered there finite sums of more general operators than in Example 4 defined on a Banach space of continuous functions and their restrictions to suitable closed cones. The assumptions of our results are satisfied also for these mappings and therefore also for a special case of cone-linear Perron-Frobenius operators studied there.